The Geometric P=W conjecture and Thurston's compactification

TL;DR

This paper advances the understanding of the geometric P=W conjecture for $ ext{SL}(2, ext{C})$ by constructing a new projective compactification of character varieties with toric boundary divisors and a spherical dual intersection complex.

Contribution

It introduces a novel projective compactification of $ ext{SL}(2, ext{C})$-character varieties with explicit boundary structure and a new formula for multicurve embeddings.

Findings

Constructed a projective compactification with toric boundary divisors.

Established the dual intersection complex as a sphere.

Derived an explicit multicurve embedding formula.

Abstract

In this paper, we use new results together with established facts about Thurston's compactification of Teichm\"uller space to address the geometric P=W conjecture for $\mathrm{SL}(2,\mathbb{C})$, which concerns projective compactifications of character varieties of closed surfaces. In particular, we construct a projective compactification of the $\mathrm{SL}(2,\mathbb{C})$-character variety of any closed surface of genus $g>1$, in which the boundary divisors are toric varieties and the dual intersection complex is a sphere. A main technical step, of independent interest, is the derivation of an explicit formula for a well-known embedding of the set of isotopy classes of multicurves on a closed surface of genus $g$ into $\mathbb{N}^{9g-9}$.